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How a Doodler's Problem Sparked a Controversy in Math | Scientific American
(www.scientificamerican.com)
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this post was submitted on 28 Jul 2023
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Are their exotic 2D geometries that work? Like, for example, the surface of a Klein bottle? (A cursory google search rules out the Klein bottle -- any others perhaps?)
Not for contiguous spaces. Think of a triangle surrounded by three rectangles. There's no way for all three outer rectangles to touch without enveloping the center triangle. There are only two directions you can go, and as long as the center triangle is enveloped, it cannot connect to a fifth shape without separating two of the outer shapes.
Alright, non Euclidean maps it is then!
On a torus, you can have up to seven mutually adjacent regions. See https://upload.wikimedia.org/wikipedia/commons/3/37/Projection_color_torus.png
Oh, very nice!